Se ha denunciado esta presentación.
Utilizamos tu perfil de LinkedIn y tus datos de actividad para personalizar los anuncios y mostrarte publicidad más relevante. Puedes cambiar tus preferencias de publicidad en cualquier momento.
Integrales
www.fisicaeingenieria.es
En este document encontrarás toda la metodología para resolver cualquier tipo de integ...
www.fisicaeingenieria.es Tabla de integrales
1
Tipos FormasFormasFormasFormas
SimpleSimpleSimpleSimple CompuestaCompuestaC...
www.fisicaeingenieria.es Tabla de integrales
2
TTTTipo potencialipo potencialipo potencialipo potencial ( )1a ≠ −
0dx C=∫ ...
www.fisicaeingenieria.es Tabla de integrales
3
1
dx L x
x
=∫
f
dx L f
f
′
=∫
Ejercicios resueltos:Ejercicios resueltos:Eje...
www.fisicaeingenieria.es Tabla de integrales
4
Tipo senoTipo senoTipo senoTipo seno
s n cose xdx x=−∫
s n cose f f dx f′⋅ ...
www.fisicaeingenieria.es Tabla de integrales
5
Tipo tangenteTipo tangenteTipo tangenteTipo tangente
2
sec tanxdx x=∫
2
sec...
www.fisicaeingenieria.es Tabla de integrales
6
Tipo arco seno (=arco coseno)Tipo arco seno (=arco coseno)Tipo arco seno (=...
www.fisicaeingenieria.es Tabla de integrales
7
55.55.55.55.
( )
2
24 2
1 2 1
arctg
1 2 21
x x
dx dx x
x x
= =
+ +
∫ ∫
56.5...
www.fisicaeingenieria.es Tabla de integrales
8
6.6.6.6.
( )
( ) ( )
( )
4 1
4 4
4
2 11 1 1
2 1 2 1 2
2 2 4 12 1
x
dx x dx ...
www.fisicaeingenieria.es Tabla de integrales
9
2
2
1 2 2
1 2
2
cos x sen x
cos x
sen x
− =
−
=
13.13.13.13. ( ) ( )2 2 2
1...
www.fisicaeingenieria.es Tabla de integrales
10
20.20.20.20. ( )
3
2 23
2 2 2 2 2
2
1
1 1 2 1
31 2
xx
dx x x x x dx x x C
...
www.fisicaeingenieria.es Tabla de integrales
11
25.25.25.25. 2 2 22
1 1
1 1 1 1 14 22
44 4 4 2 2
1 1
4 4 2 2
x
dx dx dx dx...
www.fisicaeingenieria.es Tabla de integrales
12
Resumiendo: 2
2
x x x x
x x x
x x
x
e cosx dx e cosx e senx e cosx dx
e co...
www.fisicaeingenieria.es Tabla de integrales
13
39.39.39.39.
( ) ( ) 22
1 1 1
2 2 2 2
111
dx t dt dt arctgt C arctg x C
tt...
www.fisicaeingenieria.es Tabla de integrales
14
1 0 0 1
1
1 1 1
0
−
1 1 1
( )( )2 2
1 1 1t t t t− = − + +
1 1 1 0 0 0 0
1
...
www.fisicaeingenieria.es Tabla de integrales
15
2
1
2 1 2 1
1 2
t
t dt t ln t C
t
  
= − + = − + + + =  
+   
...
www.fisicaeingenieria.es Tabla de integrales
16
2 2
2 2
2
1
cos x cos x sen x
cos x sen x
= −
= +
2
2
1 2 2
1 2
2
cos x co...
www.fisicaeingenieria.es Tabla de integrales
17
60.60.60.60.
2 3
2 1
2 2 2 1
1 1 11
x t t
dx t dt dt t t dt
t t tx
 
= ⋅...
www.fisicaeingenieria.es Tabla de integrales
18
1 1 1
3
9 3 9
x
x
x ln e C
e
= − + + − +
63.63.63.63.
( ) ( )
( ) 22
1 1 1...
www.fisicaeingenieria.es Tabla de integrales
19
4
4 2
2
2
3
3 12
12
48
x
x x
x
x
− +
−12 +
48
( ) ( )
( )( )
( ) ( )
2
2
4...
www.fisicaeingenieria.es Tabla de integrales
20
( ) ( ) 2
2 1 1 1
0 1 1
1 3 3
2 5 2 4 5 2 1 12 6 2 3
x Ax x B x Cx
x B B
x...
www.fisicaeingenieria.es Tabla de integrales
21
2
2 2
1 2 2 2
2 2 2 2 2
sen t sent cost
cos t t t
cos t dt dt C C
⋅
+
= ⋅ ...
www.fisicaeingenieria.es Tabla de integrales
22
1 2 1 4 2 4
2 8 2 4 8 32
cos x cos x x sen x x sen x
dx C
− − 
= − = − −...
www.fisicaeingenieria.es Tabla de integrales
23
( )
2
3 3 3
3
1 3
x x
x senx ln cosx
cosx dx C
ln
+ ⋅
⋅ ⋅ = +
+
∫
82.82.82...
www.fisicaeingenieria.es Tabla de integrales
24
1 1 2 0
0
0 0 0
1 1 2 0
1
1 2
1 2 0
2
2
0
−
−
−
−
1
( )( )
( )( ) ( ) ( )
...
www.fisicaeingenieria.es Tabla de integrales
25
89.89.89.89.
2 2
2 2 2 2 2 2 2
2
a sen t a a
a b x dx a b cost dt a a sen ...
www.fisicaeingenieria.es Tabla de integrales
26
1 41 11 1
3 32 21 11 1
3 32 2
5 4 7 5 4 7
1 1 4 11 1
3 2 3 2
x x x x
C C
+...
Próxima SlideShare
Cargando en…5
×

Cuaderno+de+integrales

252 visualizaciones

Publicado el

  • Sé el primero en comentar

  • Sé el primero en recomendar esto

Cuaderno+de+integrales

  1. 1. Integrales www.fisicaeingenieria.es En este document encontrarás toda la metodología para resolver cualquier tipo de integral definido o indefinida Luis Muñoz Mato
  2. 2. www.fisicaeingenieria.es Tabla de integrales 1 Tipos FormasFormasFormasFormas SimpleSimpleSimpleSimple CompuestaCompuestaCompuestaCompuesta Tipo potencial a ≠ -1 ∫ + = + 1 1 a x dxx a a ∫ + =′⋅ + 1 1 a f dxff a a 4 51 5 x dx x=∫ ( )( ) ( ) 312 302 1 2 1 1 31 x x x x x dx + + + + + =∫ Tipo logarítmico xLdx x =∫ 1 fLdx f f = ′ ∫ 3 1 3 3dx dx L x x x = =∫ ∫ 2 3 3 3 1 8 8 3 = + +∫ x dx L x x Tipo exponencial xx edxe =∫ La a dxa x x =∫ ∫ ′=′⋅ edxfe f La a dxfa f ′ =′⋅∫ 2 1 2 1 2 11 1 2 2 x x x e dx e dx e+ + + = ⋅ =∫ ∫ 2 1 2 1 2 11 1 3 3 2 2 ln 3 + + + = ⋅ =∫ ∫ x x x e dx dx Tipo seno ∫ = senxxdxcos ∫ =′⋅ senfdxffcos 1 1 s n 2 s n 2 2 cos 2 2 2 e x dx e x dx x= ⋅ = −∫ ∫ Tipo coseno ∫ −= xsenxdx cos ∫ −=′⋅ fdxfsenf cos ( ) ( ) ( )2 2 2 1 cos 1 s n 1x x x dx e x x+ ⋅ + + = + +∫ Tipo tangente tgxxdx =∫ 2 sec ( ) tgxdxxtg =+∫ 2 1 tgxdx x =∫ 2 cos 1 tgfdxff =′⋅∫ 2 sec ( ) tgfdxfftg =′⋅+∫ 2 1 tgfdx f f = ′ ∫ 2 cos 2 2 3sec 3 sec 3tanx dx x dx x= =∫ ∫ 2 2 7 7 sec 7 tan cos = =∫ ∫dx x dx x x ( ) ( )2 2 5 5tan 5 1 tan 5tanx dx x dx x+ = + =∫ ∫ Tipo arco seno arcsenxdx x = − ∫ 2 1 1 a x arcsendx xa = − ∫ 22 1 arcsenfdx f f = − ′ ∫ 2 1 a f arcsendx fa f = − ′ ∫ 22 ( ) 2 4 22 2 2 arcsen 1 1 x x dx dx x x x = = − − ∫ ∫ Tipo arco tangente arctgxdx x = +∫ 2 1 1 a x arctg a dx xa 11 22 = +∫ arctgfdx f f = + ′ ∫ 2 1 a f arctg a dx fa f 1 22 = + ′ ∫ 2 2 1 1 1 1 arctg 3 3 3 1 3 dx dx x x x = = + +∫ ∫ TTTT
  3. 3. www.fisicaeingenieria.es Tabla de integrales 2 TTTTipo potencialipo potencialipo potencialipo potencial ( )1a ≠ − 0dx C=∫ k dx kx C= +∫ 1 1 a a x x dx a + = +∫ 1 1 a a f f f dx a + ′⋅ = +∫ Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos: 1.1.1.1. 4 51 5 x dx x=∫ 2.2.2.2. 3 3 4 4 1 3 3 x x dx x dx x − − − = = = − −∫ ∫ 3.3.3.3. 5 2 53 3 3 3 5 5 3 x x dx x= =∫ 4.4.4.4. 2 1 23 3 3 3 2 2 3 x x dx x − = =∫ 5.5.5.5. ( ) ( ) 2 31 1 1 3 x dx x+ = +∫ 6.6.6.6. ( )( ) ( ) 312 302 1 2 1 1 31 x x x x x dx + + + + + =∫ 7.7.7.7. 3 41 s n cos s n 4 e x x dx e x=∫ 8.8.8.8. ( ) 2 2 31 sec 3 tg x x dx tg x=∫ 9.9.9.9. ( ) ( )3 5 3 2 41 1 4 tg x tg x dx tg x tg x dx tg x+ = + =∫ ∫ 10.10.10.10. ( )3 3 2 4 31 1 cos 1 s n sin 4 3 xdx tg x tg x dx tg x e x x= + = = −∫ ∫ 11.11.11.11. ( ) ( )3 2 2 31 s n 1 cos s n cos s n cos cos 3 sen xdx e x x dx e x e x dx x x= − = − = − +∫ ∫ ∫ Tipo logarítmicoTipo logarítmicoTipo logarítmicoTipo logarítmico
  4. 4. www.fisicaeingenieria.es Tabla de integrales 3 1 dx L x x =∫ f dx L f f ′ =∫ Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos: 1.1.1.1. 3 1 3 3dx dx L x x x = =∫ ∫ 2.2.2.2. 2 3 3 3 1 5 5 x dx L x x x x + = + + + +∫ 3.3.3.3. ( )2 2 2 1 2 1 1 1 2 1 2 x x dx dx L x x x = = + + +∫ ∫ 4.4.4.4. 2 2 3 3 3 1 3 1 8 8 3 8 3 x x dx dx L x x x = = + + +∫ ∫ 5.5.5.5. s n cos cos e x tg x dx dx Ln x x = = −∫ ∫ 6.6.6.6. cos cotg s n s n x x dx dx L e x e x = =∫ ∫ Tipo exponencialTipo exponencialTipo exponencialTipo exponencial x x e dx e=∫ x x a a dx La =∫ f f e f dx e′⋅ =∫ f f a a f dx La ′⋅ =∫ Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos: 12.12.12.12. 2 1 2 1 2 11 1 2 2 x x x e dx e dx e+ + + = ⋅ =∫ ∫ 13.13.13.13. 3 3 3 x x dx L =∫ 14.14.14.14. 3 3 3 2 32 2 2 x xx x dx dx L        = =         ∫ ∫ 15.15.15.15. 2 2 21 1 2 2 2 x x x x e dx x e dx e= ⋅ =∫ ∫ 16.16.16.16. sin sin cosx x e x dx e=∫ 17.17.17.17. 2 2 s n s n 2s n cos s ne x e x e e x x dx e e x dx e= =∫ ∫
  5. 5. www.fisicaeingenieria.es Tabla de integrales 4 Tipo senoTipo senoTipo senoTipo seno s n cose xdx x=−∫ s n cose f f dx f′⋅ = −∫ Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos: 18.18.18.18. 1 1 s n 2 s n 2 2 cos2 2 2 e x dx e x dx x= ⋅ = −∫ ∫ 19.19.19.19. ( ) ( ) ( ) 1 1 s n 2 6 s n 2 6 2 cos 2 6 2 2 e x dx e x dx x+ = + ⋅ = − +∫ ∫ 20.20.20.20. ( ) ( ) ( )2 2 21 1 s n 3 s n 2 cos 3 2 2 x e x dx e x x dx x⋅ + = ⋅ = − +∫ ∫ 21.21.21.21. ( ) ( ) ( )2 2 2 1 s n 1 cos 1x e x x dx x x+ ⋅ + + = − + +∫ 22.22.22.22. ( ) ( ) ( s n 1 s n cos e Lx dx e Lx dx L x x = ⋅ = −∫ ∫ 23.23.23.23. ( ) ( )s n cosx x x e e e dx e= −∫ 24.24.24.24. ( ) 1 1 c s n5 s n5 5 cos5 5 5 e x dx e x dx x= ⋅ = − =−∫ ∫ 25.25.25.25. ( ) ( ) ( ) 1 1 s n 7 8 s n 7 8 7 cos 7 8 7 7 e x dx e x dx x+ = + ⋅ =− +∫ ∫ Tipo cosenoTipo cosenoTipo cosenoTipo coseno cos s nxdx e x=∫ cos s nf f dx e f′⋅ =∫ Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos: 26.26.26.26. 1 1 cos2 cos2 2 sin 2 2 2 x dx x dx x= ⋅ =∫ ∫ 27.27.27.27. ( ) ( ) ( ) 1 1 cos 2 1 cos 2 1 2 s n 2 1 2 2 x dx x dx e x+ = + ⋅ = +∫ ∫ 28.28.28.28. ( ) ( )2 21 cos 1 cos 2 x x dx x⋅ + =∫ ∫ 29.29.29.29. ( ) ( ) ( )2 2 2 1 cos 1 s n 1x x x dx e x x+ ⋅ + + = + +∫ 30.30.30.30. ( ) ( ) ( ) cos 1 cos s n Lx dx Lx dx e Lx x x = ⋅ =∫ ∫ 31.31.31.31. cos s nx x x e e dx e e=∫ 32.32.32.32. ( ) ( )2 3 3 2 3 3 cos 9 cos 9 3 s nx x dx x x dx e x+ = + ⋅ = +∫ ∫ ∫ 33.33.33.33. ( ) ( ) ( )2 3 3 2 31 1 cos 1 cos 1 3 s n 1 3 3 x x dx x xdx e x+ = + ⋅ = +∫ ∫
  6. 6. www.fisicaeingenieria.es Tabla de integrales 5 Tipo tangenteTipo tangenteTipo tangenteTipo tangente 2 sec tanxdx x=∫ 2 sec tanf f dx f′⋅ =∫ Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos: 34.34.34.34. 2 2 3sec 3 sec 3tanx dx x dx x= =∫ ∫ 35.35.35.35. 2 2 2 7 7sec 7 sec 7 tan cos dx x dx x dx x x = = =∫ ∫ ∫ 36.36.36.36. ( ) ( )2 2 5 5tan 5 1 tan 5tanx dx x dx x+ = + =∫ ∫ 37.37.37.37. ( ) ( ) ( )2 2 3 2 3 2 3 9 3 sec 9 sec 9 3 tanx x dx x x dx x + + = + ⋅ =∫ ∫ 38.38.38.38. ( ) ( ) ( )2 21 1 sec 2 1 sec 2 1 2 tan 2 1 2 2 x dx x dx x+ = + ⋅ = +∫ ∫ 39.39.39.39. ( ) ( )4 2 2 2 2 2 31 sec 1 tan sec sec tan sec tan tan 3 x dx x x dx x x x x x dx= + = + = +∫ ∫ ∫ 40.40.40.40. ( )2 2 tan 1 tan 1 tanx dx x dx x x = + − = − ∫ ∫ Tipo cotangenteTipo cotangenteTipo cotangenteTipo cotangente 2 cosec cotgx dx x=−∫ 2 cosec cotgf f dx f′⋅ =−∫ Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos: 41.41.41.41. 2 2 3 cosec 3 cosec 3 cotgx dx x dx x= = −∫ ∫ 42.42.42.42. 2 2 2 8 8 cosec 8 cosec 8 cotg sin dx x dx x dx x x = = = −∫ ∫ ∫ 43.43.43.43. ( ) ( )2 2 5 5 cotg 5 1 cotg 5 cotgx dx x dx x+ = + = −∫ ∫ 44.44.44.44. ( ) ( ) ( )2 2 1 cosec 2 1 cosec 2 1 2 cotg 2 1 2 x dx x dx x+ = + ⋅ = − +∫ ∫ 45.45.45.45. ( )2 2 cotg 1 cotg 1 cotgx dx x dx x x = + − = − − ∫ ∫ 46.46.46.46. ( ) ( )4 2 2 2 2 2 cosec 1 cotg cosec cosec cotg cosecx dx x x dx x x x dx= + = + =∫ ∫ ∫ 31 cotg cotg 3 x x dx− −
  7. 7. www.fisicaeingenieria.es Tabla de integrales 6 Tipo arco seno (=arco coseno)Tipo arco seno (=arco coseno)Tipo arco seno (=arco coseno)Tipo arco seno (=arco coseno) 2 1 arcsen 1 = − ∫ dx x x 2 arcsen 1 ′ = − ∫ f dx f f Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos:Ejercicios resueltos: 47.47.47.47. ( ) 2 4 22 2 2 arcsen 1 1 x x dx dx x x x = = − − ∫ ∫ 48.48.48.48. ( ) 2 2 arcsen 1 1 x x x x x e e dx dx e e e = = − − ∫ ∫ 49.49.49.49. ( ) ( )2 2 1 1 arcsen 1 1 xdx dx Lx x L x Lx = = − − ∫ ∫ 50.50.50.50. ( ) 2 1 1 1 2 2arcsen 1 2 1 dx dx x x x x x = ⋅ = − − ∫ ∫ Tipo arco tangente (=Tipo arco tangente (=Tipo arco tangente (=Tipo arco tangente (=----arco cotangente)arco cotangente)arco cotangente)arco cotangente) 2 1 arctg 1 = +∫ dx x x 2 arctg 1 ′ = +∫ f dx f f EjeEjeEjeEjercicios resueltos:rcicios resueltos:rcicios resueltos:rcicios resueltos: 51.51.51.51. 2 2 1 1 1 1 arctg 3 3 3 1 3 dx dx x x x = = + +∫ ∫ 52.52.52.52. ( ) 22 1 1 3 1 arctg 3 1 9 3 31 3 dx dx x x x = = + + ∫ ∫ 53.53.53.53. ( ) ( ) 2 3 23 3 1 arctg 2 1 2 x dx x x x x + = + + + + + ∫ 54.54.54.54. ( )2 cos arctg sin 1 sin x dx x x = +∫
  8. 8. www.fisicaeingenieria.es Tabla de integrales 7 55.55.55.55. ( ) 2 24 2 1 2 1 arctg 1 2 21 x x dx dx x x x = = + + ∫ ∫ 56.56.56.56. ( ) 2 2 3 26 3 1 3 1 arctg 1 3 31 x x dx dx x x x = = + + ∫ ∫ 57.57.57.57. ( ) 22 arctg 1 1 x x x x x e e dx dx e e e = = + + ∫ ∫ INTEGRALESINTEGRALESINTEGRALESINTEGRALES INDEFINIDASINDEFINIDASINDEFINIDASINDEFINIDAS 1.1.1.1. ( ) 4 3 2 3 2 5 4 4 2 2 x x x x x x dx C+ + − = + − +∫ 2.2.2.2. 10 10 10 x x dx C ln = +∫ 3.3.3.3. ( )2 2 2 2 2 2x x x x x e dx x e x x dx x e x e dx⋅ = ⋅ − ⋅ ⋅ = ⋅ − ⋅ ⋅ = ∗∫ ∫ ∫ 2 2 x x u x du xdx dv e dx v e = ⇒ = = ⇒ = x x u x du dx dv e dx v e = ⇒ = = ⇒ = ( ) 2 2 2 2x x x x x x x e x e e dx x e x e e C   ∗ = ⋅ − ⋅ − = ⋅ − ⋅ − + =  ∫ ( )2 2 2 2 2 2x x x x x e xe e C e x x C= ⋅ − + + = − + + 4.4.4.4. ( ) ( ) 2 1 1 2 x x x e e e dx C + + = +∫ 5.5.5.5. ( ) 1 1 3 3 3 2 23 1 1 3 1 3 4 4 x x e x dx e dx x dx x dx dx dx x x x xx −− −  + − − + = + − − + =    ∫ ∫ ∫ ∫ ∫ ( ) 1 1 3 1 23 1 4 4 3 1 41 3 x x e x dx ln x x dx + −− − = + − ⋅ ⋅ − + = + ∫ ∫ ( ) 14 1 2 133 41 3 4 14 2 11 3 3 x xx x e ln x C − + − + − = + − ⋅ − + ⋅ + = − − ++ ( ) 4 2 3 3 3 3 3 4 4 8 x e x x ln x C x − − + ⋅ − − − +
  9. 9. www.fisicaeingenieria.es Tabla de integrales 8 6.6.6.6. ( ) ( ) ( ) ( ) 4 1 4 4 4 2 11 1 1 2 1 2 1 2 2 2 4 12 1 x dx x dx x dx C x − + − − + = + = + ⋅ ⋅ = ⋅ + − ++ ∫ ∫ ∫ ( ) ( ) 3 3 1 1 2 1 6 6 2 1 x C C x − = − ⋅ + + = − + + 7.7.7.7. 2 3 3 7 5 5 4 7 5 3 4 7 3 x dx ln x x C x x +   = + + +     + +    ∫ 8.8.8.8. ( ) ( ) ( ) ( ) ( ) 4 12 42 4 32 2 2 1 1 2 1 4 1 3 x xx dx x x x dx C C x x x x + ++ = + ⋅ + = + = − + − ++ + ∫ ∫ 9.9.9.9. ( )2 x dx x tgx tgx dx x tgx ln cosx C x tgx ln cosx C cos x = ⋅ − ⋅ = ⋅ − − + = ⋅ + +∫ ∫ 2 1 u x du dx v tgx dv dx cos x = ⇒ = = ⇒ = 10.10.10.10. 1 1 1 1 1 1x x x x dx dx dt e e te t e t − = = ⋅ = + + + ∫ ∫ ∫ 1 x e t x lnt dx tdt = ⇒ = = 1 x lne lnt xlne lnt lne x lnt = = ⇒ = = 2 2 1 1 1 1 1 x dt dt arctgt C arctge C t t t t ⋅ = = + = + + +∫ ∫ 11.11.11.11. ( ) ( )2 3 2 2 3 2 31 1 1 1 3 3 3 tg x x dx tg x x dx tgx C+ ⋅ ⋅ = + ⋅ ⋅ = +∫ ∫ 12.12.12.12. ( )2 1 2 1 1 2 1 2 2 2 2 2 cos x sen x sen xdx dx cos x dx x C ⋅   = = − = − +    ∫ ∫ ∫ 2 2 2 2 2 1 cos x cos x sen x cos x sen x − = − + = + 1 2 2 2 2 2 2 x senx cosx C x senx cosx C ⋅ ⋅ = − + ⋅ = − +
  10. 10. www.fisicaeingenieria.es Tabla de integrales 9 2 2 1 2 2 1 2 2 cos x sen x cos x sen x − = − = 13.13.13.13. ( ) ( )2 2 2 1 1 1tg xdx tg x dx tg x dx dx tgx x C= + − = + − = − +∫ ∫ ∫ ∫ 14.14.14.14. ( ) ( ) ( )2 2 2 2 1 1 1tg x dx tg x dx dx tg x dx x tgx C+ = + + = + + = + +∫ ∫ ∫ ∫ 15.15.15.15. 2 2 2 1 1 2 2 2 2 x x x x lnx dx lnx dx lnx xdx x ⋅ ⋅ = ⋅ − ⋅ = ⋅ − =∫ ∫ ∫ 2 2 2 2 1 2 2 2 2 4 x x x x lnx C lnx C= − ⋅ + = − + 2 1 2 u lnx du dx x x dv xdx v = ⇒ = = ⇒ = 16.16.16.16. ( ) ( ) 1 12 2 1 2 2 2 11 1 1 1 2 12 2 1 2 x x x dx x x dx C + + + ⋅ ⋅ = + ⋅ ⋅ = + = + ∫ ∫ ( ) ( ) 3 3 2 22 2 1 11 32 3 2 x x C C + + = + = + 17.17.17.17. ( ) ( ) 1 1 21 2 1 1 2 13 3 1 3 1 senxcosx dx senx cosx dx C senx − + − − +  +  = + ⋅ ⋅ = + =  +   ∫ ∫ ( ) 1 2 1 3 6 1 1 2 senx C senx C + = + = + + 18.18.18.18. ( ) 4 4 3 31 1 1 3 x x x x x x x x x x x x e e e e e dx dx e e dx x e C e e e e − − + + = + + = + + = + − +    ∫ ∫ ∫ 19.19.19.19. 2 2 2 1 1 1 13 3 3 39 9 1 9 9 3 x dx dx dx arcsen C x x x   = = = +   −  − −    ∫ ∫ ∫
  11. 11. www.fisicaeingenieria.es Tabla de integrales 10 20.20.20.20. ( ) 3 2 23 2 2 2 2 2 2 1 1 1 2 1 31 2 xx dx x x x x dx x x C x + = ⋅ + − + ⋅ ⋅ = ⋅ + − + = + ∫ ∫ 21.21.21.21. 2 2 2 2 1 1 u x du xdx x dv dx v x x = ⇒ = = ⇒ = + + 22.22.22.22. 2 2 1 1 1 1 1 1 1lnx dx lnx dx lnx dx lnx C x x x x x x x x   = − − − = − − − = − − +    ∫ ∫ ∫ 2 1 1 1 u lnx du dx x dv dx v x x = ⇒ = = ⇒ = − 23.23.23.23. ( ) ( ) ( ) ( ) 2 2 2 22 3 2 3 5 5 1 5 5 4 4 x x dx dx cos x cos xsen x sen x    + = + =     ∫ ∫ ∫ ( ) ( ) ( ) 2 3 22 3 5 3 5 4 5 5 4 3 4 4 3 4 x dx dx cotg x tg x C cos xsen x − = − + = − + +∫ ∫ 24.24.24.24. ( ) ( ) ( ) ( ) ( )2 5 2 5 2 5 2 5 2 5 1 1 1 1 2 2 2 2 x x x x x dx e dx e dx e C C e e − + − + − + + + = = − =− + = − + −∫ ∫ ∫ ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 3 2 1 1 3 2 2 1 1 3 3 x x x x C x x x C x x x x C C = ⋅ + − + ⋅ + + =   = − + + + =   − − + = + + = +
  12. 12. www.fisicaeingenieria.es Tabla de integrales 11 25.25.25.25. 2 2 22 1 1 1 1 1 1 14 22 44 4 4 2 2 1 1 4 4 2 2 x dx dx dx dx arctg C xx x x   = = = ⋅ = +  +     + + +        ∫ ∫ ∫ ∫ 26.26.26.26. 2 2 2 3 3 3 3 2 2 3 2 2 4 4 4 3 4 3 x x x dx dx dx ln x C x x x = = = + + + + +∫ ∫ ∫ 27.27.27.27. ( ) 2 2 24 2 1 1 1 1 1 1 1 2 2 1 2 21 x x x e dx dt dt arctg t C arctg e C e tt = ⋅ = = + = + + ++ ∫ ∫ ∫ 28.28.28.28. ( ) 2 2 3 3 6x x e x dx e C− − ⋅ − = +∫ 29.29.29.29. 2 2 2 1 2 tgx tg x dx tgx dx C cos x cos x = ⋅ = +∫ ∫ Otra forma de hacerla: ( ) ( ) ( ) 2 3 3 2 1 2 2 cosxsenx dx cosx senx dx C C cos x cos x − − = − − = − + = + −∫ ∫ 30.30.30.30. tgx dx ln cosx C⋅ = − +∫ 31.31.31.31. ( ) ( ) ( ) ( ) 7 3 4 7 3 4 1 1 7 7 3 4 4 7 4 1 3 7 3 4 7 4 7 4 7 4 cos x sen x dx cos x dx sen x dx cos x dx sen x dx sen x cos x sen x cos x C C − = ⋅ − ⋅ = = ⋅ ⋅ − ⋅ = ⋅ ⋅ = = − − + = + + ∫ ∫ ∫ ∫ ∫ 32.32.32.32. ( ) ( ) ( )2 22 2 2 1 2 1 4 2 21 4 1 4 x x dx dx arctg x C x x = = + + + + + + ∫ ∫ 33.33.33.33. ( ) 2 1 2 lnxlnx dx lnx dx C x x = ⋅ = +∫ ∫ 34.34.34.34. x x x x x x e cosx dx e cosx e senx dx e cosx e senx e cosx dx ⋅ ⋅ = ⋅ + ⋅ ⋅ = ⋅ + ⋅ − ⋅ ⋅  ∫ ∫ ∫ x x u cosx du senxdx dv e dx v e = ⇒ = − = ⇒ = x x u senx du cosxdx dv e dx v e = ⇒ = = ⇒ =
  13. 13. www.fisicaeingenieria.es Tabla de integrales 12 Resumiendo: 2 2 x x x x x x x x x x e cosx dx e cosx e senx e cosx dx e cosx dx e cosx e senx e cosx e senx e cosx dx C ⋅ = + − ⋅ = ⋅ = + = + ⋅ = + ∫ ∫ ∫ ∫ 35.35.35.35. ( )4 2 2 2 2 1sen x dx sen x sen x dx sen x cos x dx⋅ = ⋅ ⋅ = − =∫ ∫ ∫ ( ) 2 2 2 2 2 2 4 sen x sen x sen x cos x dx sen x dx   − ⋅ = − =    ∫ ∫ 2 2 2 2 2 2 2 1 1 2 2 1 2 2 cos x cos x sen x cos x sen x cos x sen x cos x sen x − = − + = + − = − = 1 4 1 2 2 2 4 1 2 1 4 2 2 8 8 cos x cos x dx cos x cos x dx −   − = − =        = − − + =    ∫ ∫ 2 4 3 2 4 3 2 4 8 2 8 8 2 8 sen x sen x cos x cos x dx x C   − + = − + +  −  ∫ 3 2 4 8 4 32 sen x sen x x C= − + + 36.36.36.36. senx senx e cosx dx e C⋅ ⋅ = +∫ 37.37.37.37. ( )3 3 3 2 3 2 1sen x cos x dx sen x cos x cosx dx sen x sen x cosx dx⋅ ⋅ = ⋅ ⋅ ⋅ = − ⋅ ⋅ =∫ ∫ ∫ ( ) 4 6 3 5 4 6 sen x sen x sen x cosx sen x cosx dx C= ⋅ − ⋅ = − +∫ Otra forma de hacerlo: ( )2 3 2 3 1senx sen x cos x dx senx cos x cos x dx⋅ ⋅ ⋅ = − ⋅ ⋅ =∫ ∫ ( ) ( )3 3 3 5 4 6 4 6 cos x senx dx cos x senx dx cos x senx dx cos x senx dx cos x cos x C ⋅ ⋅ − ⋅ ⋅ = − − + − = = − + + ∫ ∫ ∫ ∫ 38.38.38.38. ( ) ( ) ( ) ( )2 2 21 1 1 1 2 1 2 2 x cos x dx cos x x dx sen x C   ⋅ + = − + ⋅ − ⋅ = − + +   ∫ ∫
  14. 14. www.fisicaeingenieria.es Tabla de integrales 13 39.39.39.39. ( ) ( ) 22 1 1 1 2 2 2 2 111 dx t dt dt arctgt C arctg x C tt tx x = ⋅ ⋅ = = + = + +++ ∫ ∫ ∫ 2 2 x t x t dx tdt = ⇒ = = 40.40.40.40. 2 1 4 5 1 43 3 3 3 9 3 3 3 3 x dx dx ln x ln x C x x x  −−  = + = − − + + + − − +    ∫ ∫ 1 0 9 3 3 9 1 3 0 3 3 1 0 − − ( ) ( ) ( )( ) ( ) ( ) 2 2 5 9 3 3 3 35 9 3 3 5 3 3 13 2 6 3 43 8 6 3 x A B x x x A x B xx x x x x A x B x x A A x B B − = + − − + + + −− = − − + − = + + − = ⇒ − = ⇒ = − = − ⇒ − = − ⇒ = 41.41.41.41. 3 3 3 5 5 5 x x x x x e e dx dx ln e C e e = = + + + +∫ ∫ 42.42.42.42. ( ) ( ) ( )3 3 4 36 3 5 5 6 4 24 223 1 1 6 6 6 11 t t t tx x t t dx t dt t dt dt t t tt tx x − −− − = ⋅ = ⋅ ⋅ = = − −−− ∫ ∫ ∫ ∫ 6 6 5 6 x t t x dx t dt = ⇒ = = ( )( ) ( )( ) 4 2 6 5 4 6 3 2 6 4 3 2 4 3 26 6 6 6 6 1 1 6 6 1 1 1 1 6 1 1 3 2 3 6 6 1 2 3 2 3 6 6 1 2 t t t t t t t dt dt t t t t t t t dt t t t t t t ln t C x x x x x ln x C − + + + = = = − + +   = + − + − + =  +  = + − + − + + + = = + − + − + + + ∫ ∫ ∫
  15. 15. www.fisicaeingenieria.es Tabla de integrales 14 1 0 0 1 1 1 1 1 0 − 1 1 1 ( )( )2 2 1 1 1t t t t− = − + + 1 1 1 0 0 0 0 1 1 0 1 1 1 1 1 − − − − − 1 0 1 −1 −1 −1 43.43.43.43. ( ) 2 2 2 1 1 1 1 1 11 1 1 1 x x x x x dx dx dx dx xx x x x + + ⋅ − − − = = = = −− − ⋅ − − ∫ ∫ ∫ ∫ 2 2 1 1 x sent t arcsenx dx cost cost sen t x = ⇒ = = = − = − 2 2 1 1 1 1 sen t cos t cost cost dt cost dt cost dt sent sent sent − = ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ = − − −∫ ∫ ∫ ( )( )2 2 1 11 1 1 1 sent sentcos t sen t cost dt dt dt sent sent sent − +− ⋅ ⋅ = = = − − −∫ ∫ ∫ ( ) 2 1 1sent dt t cost C arcsenx x C+ = − + = − − +∫ 44.44.44.44. ( ) ( )( ) 2 2 1 11 1 1 1 1 1 1 senx senx senx dx dx dx dx senx senx senx sen x cos x ⋅ − + + = = = = − − + −∫ ∫ ∫ ∫ 2 2 2 2 1 1senx dx dx cos x senx dx cos x cos x cos x −  + = + ⋅ ⋅ =    ∫ ∫ ∫ ( ) ( ) ( ) 1 2 2 1 1 1 cosx dx cosx senx dx tgx C tgx C cos x cosx − − = − ⋅ − = − + = + + −∫ ∫ 45.45.45.45. ( ) 2 3 2 2 2 2 2 1 1 x t t t dx t dt dt dt t t t t tx x = ⋅ ⋅ = = = + + ++ ∫ ∫ ∫ ∫ 2 2 x t x t dx tdt = ⇒ = = 1 0 0 1 1 1 1 − − 1 −1 Cociente t=1
  16. 16. www.fisicaeingenieria.es Tabla de integrales 15 2 1 2 1 2 1 1 2 t t dt t ln t C t    = − + = − + + + =   +    ∫ 2 2 2 1 2 2 1t t ln t C x x ln t C= − + + + = − + + + 46.46.46.46. 2 3 2 2 2 1 3 tg x tg x dx tg x dx C cos x cos x = ⋅ ⋅ = +∫ ∫ 47.47.47.47. 2 2 1tgx tgx tgxe dx e dx e C cos x cos x = ⋅ = +∫ ∫ 48.48.48.48. 2 2 2 2 5 1 5 2 1 10 1 9 5 9 5 2 9 5 2 9 5 2 x x x dx dx dx ln x C x x x ⋅ = = = + + + + +∫ ∫ ∫ 49.49.49.49. ( ) 3 2 3 28 8 1 8 8 1 1 3 x x x x dx x dx x C x + + + = + = + + +∫ ∫ 8 8 1 1 1 8 0 1 0 − − − 8 0 1 50.50.50.50. ( )2 2 22 2 2 2 1 2sen x cos x sen xsen x sen x sen x cos x dx dx dx senx cosx senx cosx senx cosx + ++ + + = = = ⋅ ⋅ ⋅∫ ∫ ∫ 2 2 2 2 sen x cos x senx cosx dx dx dx dx senx cosx senx cosx cosx senx + = + = ⋅ ⋅∫ ∫ ∫ ∫ 2ln cosx ln senx C− + + 51.51.51.51. ( ) ( ) 4 5 5 4 1 4 4 senxcosx dx senx cosx dx C C sen x sen x − − = ⋅ ⋅ = + =− + −∫ ∫ 52.52.52.52. ( ) ( ) ( ) ( ) 1 112 22 2 1 2 2 2 1 12 1 2 1 111 2 2 x xx dx x x dx C C x − + − − − = − − − = − + = − + = − +− ∫ ∫ 2 2 1 x C= − − + 53.53.53.53. ( ) 2 4 22 2 1 2 1 1 x dx x dx arcsenx C x x = ⋅ ⋅ = + − − ∫ ∫ 54.54.54.54. 2 1 2 2 2 2 4 cos x x sen x cos x dx dx C + ⋅ = = + +∫ ∫
  17. 17. www.fisicaeingenieria.es Tabla de integrales 16 2 2 2 2 2 1 cos x cos x sen x cos x sen x = − = + 2 2 1 2 2 1 2 2 cos x cos x cos x cos x + = + = 55.55.55.55. x cosx dx xsenx senx dx xsenx cosx C⋅ ⋅ = − ⋅ = + +∫ ∫ u x du dx dv cosx dx v senx = ⇒ = = ⋅ ⇒ = 56.56.56.56. 2 2 1 1 1 arcsenx dx x arcsenx x dx x arcsenx x C x ⋅ = ⋅ − ⋅ = ⋅ + + + − ∫ ∫ 2 1 1 u arcsenx du dx x dv dx v x = ⇒ = − = ⇒ = 57.57.57.57. 3 2 2 4 4 1 1 x x dx x dx x x − −  = + =  − −  ∫ ∫ 23 5 3 52 2 1 1 1 1 2 2 2 x x dx ln x ln x C x x −   = + + = − − + + + − +    ∫ ( )( )2 1 1 1x x x− = − + 3 2 3 4 1 4 x x x x x x − − − + − 3 2 2 4 4 1 1 x x x x x − − = + − − ( ) ( ) ( )( )2 2 1 14 4 1 1 1 1 1 1 A X B Xx A B x x X X x X X + + −− − = + ⇒ = − − + − − + ( ) ( )4 1 1X A X B X− = + + − 31 3 2 2 51 5 2 2 X a A X B B = ⇒ − = ⇒ = − = − ⇒ − = − ⇒ = 58.58.58.58. 2 2 2 2 1 1 1 cos x cos xdx dx dx ln tgx C senx cosxsenx cosx tg x cos x = = = + ⋅⋅∫ ∫ ∫ 59.59.59.59. ( ) ( )2 23 1 1 1 1 senx sen x senx cos xsen x dx dx dx cosx cosx cosx − = = = − − −∫ ∫ ∫ ( )( ) ( ) 1 1 1 1 senx cosx cosx dx senx cosx dx cosx − + = + = −∫ ∫ 2 2 sen x senx dx senx cosx dx cosx C= ⋅ + ⋅ ⋅ = − + +∫ ∫
  18. 18. www.fisicaeingenieria.es Tabla de integrales 17 60.60.60.60. 2 3 2 1 2 2 2 1 1 1 11 x t t dx t dt dt t t dt t t tx   = ⋅ ⋅ = = − + − =  + + ++   ∫ ∫ ∫ ∫ 2 2x t x t dx tdt= ⇒ = ⇒ = 1 0 0 0 1 1 1 1 1 − − − 1 −1 1 − 3 2 1 1 1 1 t t t t t = − + − + + 3 2 3 2 3 2 1 3 2 2 2 2 1 3 2 2 2 1 3 t t t ln t C t t t ln t C x x x ln x C   = − + − + + =    = − + − + + = = − + − + + 61.61.61.61. ( ) ( ) 2 2 2 1 1 1 2 2 2 21 4 1 2 1 2 x x x x x x dx dx ln dx ln = ⋅ ⋅ = ⋅ ⋅ ⋅ = − − − ∫ ∫ ∫ ( )1 2 2 x arcsen C ln = ⋅ + 2 2 2 2 2 x x x dt t ln dx dt dx ln = ⇒ ⋅ = ⇒ = 62.62.62.62. ( ) ( ) ( )2 1 1 1 1 3 33x dx dt dt t t t t te − = ⋅ = = − −−∫ ∫ ∫ 1x e t x lnt dx dt t = ⇒ = ⇒ = ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 3 31 1 3 3 3 3 At t B t CtA B C t t t t t t t t t − + − + = + + ⇒ = − − − − ( ) ( ) 2 1 3 3At t B t Ct= − + − + 10 1 3 3 13 1 9 9 t B B t C C = ⇒ = − ⇒ = − = ⇒ = ⇒ = 2 1 1 1 1 1 19 3 9 3 3 9 3 9 dt ln t ln t C t t t t  − −  = + + = − + + − + = −    ∫
  19. 19. www.fisicaeingenieria.es Tabla de integrales 18 1 1 1 3 9 3 9 x x x ln e C e = − + + − + 63.63.63.63. ( ) ( ) ( ) 22 1 1 1 2 2 12 1 2 1 dx t dt dt tx x t t = ⋅ − = − = + − − − −  ∫ ∫ ∫ 2 2 1 1 1 2x t x t x t dx tdt− = ⇒ − = ⇒ = − ⇒ = − 2 2 1arctgt C arctg x C= − + = − − + 64.64.64.64. ( ) ( ) ( )224 3 2 24 1 1 4 4 4 1 1 t t t tx x t t dx t dt t dt dt t t t t tx x + ++ + = = = = − − −−∫ ∫ ∫ ∫ 4 24 4x t x t dx t dt= ⇒ = ⇒ = 1 1 0 0 0 1 2 2 2 2 Resto1 2 2 2 = 3 2 2 2 2t t t⇒ + + + 4 3 4 3 3 2 22 2 4 4 2 2 2 4 2 2 1 1 1 4 3 t t t t dt t t t dt t t ln t C t t  +   = = + + + + = + + + + − +   − −    ∫ ∫ 4 3 2 34 4 48 8 4 8 8 1 4 8 8 1 3 3 t t t t ln t C x x x x ln x C= + + + + − + = + + + + − + 65.65.65.65. ( ) 3 2 3 4 334 1 1 4 4 4 11 t t dx t dt dt dt t t tt tx x = ⋅ ⋅ = = = − −−− ∫ ∫ ∫ ∫ 4 3 4x t x t dx t dt= ⇒ = ⇒ = 2 3 34 3 4 3 4 4 1 1 3 1 3 3 t dt ln t C ln x C t = = − + = − + −∫ 66.66.66.66. 4 2 3 2 2 2 3 48 48 3 12 12 4 4 4 x dx x dx x x dx x x x   = + + = + + =  − − −  ∫ ∫ ∫
  20. 20. www.fisicaeingenieria.es Tabla de integrales 19 4 4 2 2 2 3 3 12 12 48 x x x x x − + −12 + 48 ( ) ( ) ( )( ) ( ) ( ) 2 2 48 4 2 2 2 248 4 2 2 48 2 2 2 48 4 12 2 48 4 12 A B x x x A x B x x x x A x B x x A A x B B = + − − + + + − = − − + = + + − = ⇒ = ⇒ = = − ⇒ = − ⇒ = − 3 3 92 92 12 2 2 12 12 2 12 2 x x dx x x x x ln x ln x C   = + + − =  − +  = + + − − + + ∫ 67.67.67.67. ( ) 1 1 1 1 1 2 1 1 1 1 x x e t t dx dt dt dt e t t t t t t + + +   = ⋅ = = + =  − − − −  ∫ ∫ ∫ ∫ 1x e t x lnt dx dt t = ⇒ = ⇒ = ( ) ( ) ( ) ( ) 11 1 1 1 1 1 A t Btt A B t t t t t t t t t − ++ + = + ⇒ = − − − − ( ) 0 1 1 1 1 2 t A t A t Bt t B = ⇒ = + = − + ⇒ = ⇒ = 2 1 2 1 2 1x x x ln t ln t C lne ln e C x ln e C= + − + = + − + = + − + 68.68.68.68. 1 1 1 1 2 2 2 1 1 11 1 t t dx t dt dt dt t t tx + − = ⋅ ⋅ = = = + + ++ + ∫ ∫ ∫ ∫ 1 1 1 2 2 1 1 1 t dt dt dt dt t t t +    = − = − =   + + +    ∫ ∫ ∫ ∫ 2 2 1 1 1 2x t x t x t dx tdt+ = ⇒ + = ⇒ = − ⇒ = 69.69.69.69. ( )2 2 2 1 3 1 3 1 3 3 1 1 1 x dx dx ln x ln x C x x x x x x + − −  = + + = − + + − +  − −  ∫ ∫ ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 1 12 1 2 1 1 1 1 1 Ax x B x Cxx A B C x x x x x x x x x x − + − ++ + = + + ⇒ = = − − − −
  21. 21. www.fisicaeingenieria.es Tabla de integrales 20 ( ) ( ) 2 2 1 1 1 0 1 1 1 3 3 2 5 2 4 5 2 1 12 6 2 3 x Ax x B x Cx x B B x C C x A B C A A A + = − + − + = ⇒ = − ⇒ = − = ⇒ = ⇒ = = ⇒ = + + ⇒ = − + ⇒ − = ⇒ = − 70.70.70.70. 2 2 1 1 1 2 2 2 1 1 1 dx t dt dt arcsent C x x t t t = ⋅ ⋅ = = + = − − − ∫ ∫ ∫ 2arcsen x C= + 2 2x t x t dx tdt= ⇒ = ⇒ = 71.71.71.71. ( )3 2 2 1cos x dx cos x cosx dx sen x cosx dx⋅ = ⋅ ⋅ = − ⋅ ⋅ =∫ ∫ ∫ ( ) 3 2 3 sen x cosx sen x cosx dx senx C= − ⋅ = − +∫ 72.72.72.72. 3 3 3 3 33 3 3 3 3 sen x xcos x cos x xcos x x sen x dx dx C − ⋅ ⋅ = + = − + + =∫ ∫ 3 3 3 u x du dx cos x dv sen x dx v = ⇒ = = ⋅ ⇒ = − 3 3 3 9 xcos x sen x C= − + + 73.73.73.73. 2 2 1 1 2 1 2 1 x arctgx dx x arctgx x x x arctgx dx x x ⋅ = ⋅ − ⋅ = ⋅ − = + +∫ ∫ ∫ 2 1 1 u arctgx du dx x dv dx v x = ⇒ = + = ⇒ = 21 1 2 x arctgx ln x C= ⋅ − + + 74.74.74.74. 2 2 2 1 1x dx sen t cost dt cos t cost dt cost cost dt− = − ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ =∫ ∫ ∫ ∫ 2 2 1 1 x sent t arcsenx dx costdt cost sen t x = ⇒ = = ⇒ = − = − 2 2 2 2 2 1 cos t cos t sen t cos t sen t = − = + 2 2 1 2 2 1 2 2 cos t cos t cos t cos t + = + =
  22. 22. www.fisicaeingenieria.es Tabla de integrales 21 2 2 2 1 2 2 2 2 2 2 2 2 sen t sent cost cos t t t cos t dt dt C C ⋅ + = ⋅ = = + + = + + =∫ ∫ 2 1 2 2 2 2 t sent cost arcsenx x x C C ⋅ ⋅ − = + + = + + 75.75.75.75. ( )2 2 2 2 2 2 2 1a x dx a a sen t a cost dt a sen t a cost dt− = − ⋅ ⋅ ⋅ = − ⋅ ⋅ ⋅ =∫ ∫ ∫ x x a sent t arcsen dx a cost dt a = ⋅ ⇒ = ⇒ = ⋅ ⋅ 2 2 a cos t a cost dt a cost a cost dt⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ =∫ ∫ 2 2 2 21 2 2 2 2 4 cos t t sen t a cos t dt a dt a C +   = ⋅ = = + +    ∫ ∫ 2 2 2 1 2 2 2 2 x xx arcsen a at sent cost aa C a C    −  ⋅     = + + = + + =           2 2 2 2 2 x a arcsen x a xa C ⋅ ⋅ − = + + 76.76.76.76. ( ) ( ) 2 3 2 2 2 2 2 3 x x x x xe x e x dx e x xe dx x e dx+ = + + = + + ⋅ ⋅ =∫ ∫ ∫ x x u x du dx dv e dx v e = ⇒ = = ⇒ = ( ) 2 3 2 3 2 2 2 2 3 2 3 x x x x x xe x e x x e e dx xe e C= + + ⋅ − = + + − +∫ 77.77.77.77. ( )4 2 2 2 2 1sen x dx sen x sen x dx sen x cos x dx⋅ = ⋅ ⋅ = − =∫ ∫ ∫ ( )2 2 2 2 2 4 sen x sen x sen xcos x dx sen x dx   − = − =    ∫ ∫ 2 2 2 2 2 2 1 1 2 2 cos x cos x sen x cos x sen x cos x sen x − = + = + − = 21 2 2 cos x sen x − =
  23. 23. www.fisicaeingenieria.es Tabla de integrales 22 1 2 1 4 2 4 2 8 2 4 8 32 cos x cos x x sen x x sen x dx C − −  = − = − − + +    ∫ 3 2 4 8 4 32 x sen x sen x C= − + + 78.78.78.78. 2 2 2 2 2ln x dx x ln x lnx dx xln x x lnx dx ⋅ = ⋅ − ⋅ = − ⋅ − =  ∫ ∫ ∫ 2 1 2u ln x du lnx dx x dv dx v x = ⇒ = ⋅ ⋅ = ⇒ = 1 u lnx du dx x dv dx v x = ⇒ = = ⇒ = 2 2 2x ln x lnx x C= ⋅ − + + 79.79.79.79. 3 3 2 3x senx dx x cosx x cosx dx⋅ ⋅ = − ⋅ + ⋅ ⋅ =∫ ∫ 3 2 3u x du x dx dv senx dx v cosx = ⇒ = = ⋅ ⇒ = − 2 2u x du xdx dv cosx dx v senx = ⇒ = = ⋅ ⇒ = u x dv dx du senxdx v cosx = ⇒ = = ⇒ = − 3 2 3 2x cosx x senx x senx dx = − ⋅ + − ⋅ ⋅ = ∫ 3 2 3 6x cosx x senx x cosx cosx dx − ⋅ + ⋅ − − ⋅ + ⋅ =  ∫ 3 2 3 6 6x cosx x senx x cosx senx C= − ⋅ + ⋅ + ⋅ − + 80.80.80.80. 2 2 2 1 2 2 2 2 2 2 2 x x x x x x x x dx dx C ln ln ln ln ln − − − − − ⋅ ⋅ − ⋅ = − + = − + ⋅ + =∫ ∫ 2 2 2 x x u x du dx dv dx v ln − − = ⇒ = = ⋅ ⇒ = − ( ) 2 2 2 2 2 x x x C ln ln − − ⋅ = − + 81.81.81.81. 3 3 3 3 3 3 3 3 3x x x x x x cosx dx senx ln senx dx senx ln cosx ln cosx d⋅ ⋅ = ⋅ − ⋅ ⋅ = ⋅ − − ⋅ + ⋅ ⋅ ∫ ∫ ∫ 3 3 3x x u du ln dx dv cosx dx v senx = ⇒ = ⋅ ⋅ = ⋅ ⇒ = 3 3 3x x u du ln dx dv senx dx v cosx = ⇒ = ⋅ = ⋅ ⇒ = − ( ) 2 3 3 3 3 3 3x x x x cosx dx senx ln cosx ln cosx dx⋅ = + ⋅ ⋅ − ⋅ ⋅ =∫ ∫ ( ) 2 1 3 3 3 3 3x x x ln cosx dx senx ln cosx + ⋅ ⋅ = + ⋅ =  ∫
  24. 24. www.fisicaeingenieria.es Tabla de integrales 23 ( ) 2 3 3 3 3 1 3 x x x senx ln cosx cosx dx C ln + ⋅ ⋅ ⋅ = + + ∫ 82.82.82.82. 3 3 3 2 21 1 3 3 3 3 x x x x lnx dx lnx dx lnx x dx x ⋅ = ⋅ − ⋅ ⋅ = − =∫ ∫ ∫ 3 2 1 3 u lnx du dx x x dv x dx v = ⇒ = = ⇒ = 3 3 3 3 1 3 3 3 3 9 x x x x lnx C lnx C= − ⋅ + = − + 83.83.83.83. 2 2 1 1 1 2 2 2 dx dx tg x C x cos x cos x x = ⋅ ⋅ = + ⋅ ∫ ∫ 84.84.84.84. ( )2 2 1 1 1 1 1 dx dx arcsen lnx C xx ln x ln x = ⋅ ⋅ = + − − ∫ ∫ 85.85.85.85. ( ) ( ) 2 2 1 7 1 1 27 7 7 2 49 7 72 7 1 2 x x x x x dx dx ln dx ln x = ⋅ ⋅ = ⋅ ⋅ ⋅ = + + + ∫ ∫ ∫ 2 2 1 1 1 1 1 1 7 7 7 7 2 7 2 7 2 27 7 1 1 2 2 x x x x ln ln dx dx ln ln ⋅ = ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ =     + +        ∫ ∫ 2 7 2 2 7 2 7 2 7 22 x x arctg C arctg C ln ln    ⋅ = ⋅ + = ⋅ +       86.86.86.86. 2 3 2 1 1 2 5 1 2 1 12 2 2 1 2 2 1 2 2 2 x x dx dx ln x ln x ln x C x x x x x x   − + − = + + = + − − + +  + − − +    ∫ ∫
  25. 25. www.fisicaeingenieria.es Tabla de integrales 24 1 1 2 0 0 0 0 0 1 1 2 0 1 1 2 1 2 0 2 2 0 − − − − 1 ( )( ) ( )( ) ( ) ( ) ( )( ) 3 2 2 3 2 2 3 2 2 1 2 2 5 1 2 1 2 1 2 2 12 5 1 2 1 2 x x x x x x x x A B C x x x x x x A x x Bx x Cx xx x x x x x x x + − = − + + − = + + + − − + − + + + + −+ − = + − − + ( )( ) ( ) ( )2 2 5 1 1 2 2 1x x A x x Bx x Cx x+ − = − + + + + − 10 1 2 2 1 6 3 2 12 3 6 2 x A A x B B x C C = ⇒ − = − ⇒ = = ⇒ = ⇒ = = − ⇒ − = ⇒ = − 87.87.87.87. 2 2 1 1 1 1 1 1 1 1 x x e t t dx dt dt dt ln t C e t t t t −  = ⋅ = = + = − + +  + + + +  ∫ ∫ ∫ ∫ 1x e t x lnt dx dt t = ⇒ = ⇒ = 1x x e ln e C= − + + 1 0 1 1 1 − − 1 − 88.88.88.88. ( ) ( ) 5 3 2 4 2 1 1 2 2 2 5 3 t t x x dx t t tdt t t dt C   − ⋅ = + ⋅ = + = + + =    ∫ ∫ ∫ 2 2 1 1 1 2x t x t x t dx tdt− = ⇒ − = ⇒ = + ⇒ = ( ) ( ) 5 3 5 3 2 1 2 12 2 5 3 5 3 x xt t C C − − = + + = + +
  26. 26. www.fisicaeingenieria.es Tabla de integrales 25 89.89.89.89. 2 2 2 2 2 2 2 2 2 2 a sen t a a a b x dx a b cost dt a a sen t cost dt b b b − ⋅ = − ⋅ ⋅ ⋅ = − ⋅ ⋅ ⋅ =∫ ∫ ∫ 2 2 2 2 2 2 1 asent a x dx costdt b b bx b x a b x sent cost a a a bx t arcsen a = ⇒ = − = ⇒ = − =   =     ( )2 2 2 2 1 a a a sen t cost dt a cos t cost dt b b = − ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ =∫ ∫ 2 2 2 1 2 2 a a a cos t a cost cost dt cos t dt dt b b b + = ⋅ ⋅ ⋅ ⋅ = ⋅ = =∫ ∫ ∫ 2 2 2 2 2 2 4 2 2 a t a sen t a a sent cost C t C b b b b ⋅ = ⋅ + ⋅ + = ⋅ + ⋅ + = 2 2 2 2 2 2 2 a b a bx a b x arcsen x C b a b a a −  = + ⋅ ⋅ + =    2 2 2 2 2 2 b a arcsen x x a b xa C b   ⋅   − = + + 90.90.90.90. 2 1 1 x x x x e dx e ln e C e = − + + +∫ (Es la misma que la nº 87) 91.91.91.91. ( ) 4 3 2 4 3 2 3 2 2 3 2 3 1 2 3 7 7 7 4 3 2 4 3 2 x x x x x x x x x lnx dx x lnx x x     − + − ⋅ = − + − − − + −        ∫ ∫ ( ) 4 3 2 3 2 1 2 3 2 3 7 7 4 3 2 u lnx du dx x x x x dv x x x dx v x = ⇒ = = − + − ⇒ = − + − 4 3 2 3 2 2 3 2 3 7 7 4 3 2 4 3 2 x x x x x x x lnx dx     = − + − − − + − =        ∫ 4 3 2 4 3 2 2 3 2 3 7 7 4 3 2 16 9 4 x x x x x x x lnx x C     = − + − − − + − +        92.92.92.92. ( )1 1 1 1 3 3 3 2 2 4 5 5 4 7 7 x dx x x dx x − −  − = ⋅ − ⋅ ⋅ =    ∫ ∫
  27. 27. www.fisicaeingenieria.es Tabla de integrales 26 1 41 11 1 3 32 21 11 1 3 32 2 5 4 7 5 4 7 1 1 4 11 1 3 2 3 2 x x x x C C + − + − − = ⋅ − ⋅ ⋅ + = ⋅ − ⋅ ⋅ + = + − + 433 33 8 3 5 5 8 4 4 77 x x x C x x C− ⋅ + = − + = 3 33 8 7 3 8 5 5 7 4 7 4 7 x x x C x x x C− ⋅ + = − +

×